In this paper, we establish the formulas expressing the special values of the spectral zeta function $\zeta_{\rm\Delta}(n)$ of the Laplacian $\rm\Delta$ on some locally symmetric Riemannian manifold $\Gamma\backslash G/K$ in terms of the coefficients of the Laurent expansion of the corresponding Selberg zeta function. As an application, we give a numerical estimation of the first eigenvalue of $\rm\Delta$ by computing the values $\zeta_{\rm\Delta}(n)$ numerically, when $\Gamma\backslash G/K$ is a Riemann surface with $\Gamma$ being the quaternion group.
Publié le : 2005-01-14
Classification:
spectral zeta function,
Selberg's zeta function,
first eigenvalue,
11F72,
11M36,
35P15
@article{1160745823,
author = {HASHIMOTO, Yasufumi},
title = {Special values of the spectral zeta functions for locally symmetric Riemannian manifolds},
journal = {J. Math. Soc. Japan},
volume = {57},
number = {4},
year = {2005},
pages = { 217-232},
language = {en},
url = {http://dml.mathdoc.fr/item/1160745823}
}
HASHIMOTO, Yasufumi. Special values of the spectral zeta functions for locally symmetric Riemannian manifolds. J. Math. Soc. Japan, Tome 57 (2005) no. 4, pp. 217-232. http://gdmltest.u-ga.fr/item/1160745823/